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51
EVOLUTIONARY DRIFT AND EQUILIBRIUM SELECTION
, 1996
"... This paper develops an approach to equilibrium selection in game theory based on studying the equilibriating process through which equilibrium is achieved. The differential equations derived from models of interactive learning typically have stationary states that are not isolated. Instead, Nash equ ..."
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Cited by 71 (2 self)
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This paper develops an approach to equilibrium selection in game theory based on studying the equilibriating process through which equilibrium is achieved. The differential equations derived from models of interactive learning typically have stationary states that are not isolated. Instead, Nash equilibria that specify the same behavior on the equilibrium path, but different outofequilibrium behavior, appear in connected components of stationary states. The stability properties of these components often depend critically on the perturbations to which the system is subjected. We argue that it is then important to incorporate such drift into the model. A su±cient condition is provided for drift to create stationary states with strong stability properties near a component of equilibria. This result is used to derive comparative static predictions concerning common questions raised in the literature on refinements of Nash equilibrium
Computing sequential equilibria for twoplayer games
 In SODA ’06
, 2006
"... Koller, Megiddo and von Stengel showed how to efficiently compute minimax strategies for twoplayer extensiveform zerosum games with imperfect information but perfect recall using linear programming and avoiding conversion to normal form. Koller and Pfeffer pointed out that the strategies obtai ..."
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Cited by 27 (1 self)
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Koller, Megiddo and von Stengel showed how to efficiently compute minimax strategies for twoplayer extensiveform zerosum games with imperfect information but perfect recall using linear programming and avoiding conversion to normal form. Koller and Pfeffer pointed out that the strategies obtained by the algorithm are not necessarily sequentially rational and that this deficiency is often problematic for the practical applications. We show how to remove this deficiency by modifying the linear programs constructed by Koller, Megiddo and von Stengel so that pairs of strategies forming a sequential equilibrium are computed. In particular, we show that a sequential equilibrium for a twoplayer zerosum game with imperfect information but perfect recall can be found in polynomial time. In addition, the equilibrium we find is normalform perfect. Our technique generalizes to generalsum games, yielding an algorithm for such games which is likely to be prove practical, even though it is not polynomialtime. 1
Refinements of Nash equilibrium
 THE NEW PALGRAVE DICTIONARY OF ECONOMICS, 2 ND EDITION
"... This entry describes ways that the definition of an equilibrium among players’ strategies in a game can be sharpened by invoking additional criteria derived from decision theory. Refinements of John Nash’s 1950 definition aim primarily to distinguish equilibria in which implicit commitments are cr ..."
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Cited by 8 (0 self)
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This entry describes ways that the definition of an equilibrium among players’ strategies in a game can be sharpened by invoking additional criteria derived from decision theory. Refinements of John Nash’s 1950 definition aim primarily to distinguish equilibria in which implicit commitments are credible due to incentives. One group of refinements requires sequential rationality as the game progresses. Another ensures credibility by considering perturbed games in which every contingency occurs with positive probability, which has the further advantage of excluding weakly dominated strategies.
Pairwisestability and nash equilibria in network formation
 International Journal of Game Theory
"... Suppose that individual payo¤s depend on the network connecting them. Consider the following simultaneous move game of network formation: players announce independently the links they wish to form, and links are formed only under mutual consent. We provide necessary and su ¢ cient conditions on the ..."
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Cited by 7 (0 self)
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Suppose that individual payo¤s depend on the network connecting them. Consider the following simultaneous move game of network formation: players announce independently the links they wish to form, and links are formed only under mutual consent. We provide necessary and su ¢ cient conditions on the network link marginal payo¤s such that the set of pairwise stable, pairwiseNash and proper equilibrium networks coincide, where pairwise stable networks are robust to onelink deviations, while pairwiseNash networks are robust to onelink creation but multilink severance. Under these conditions, proper equilibria in pure strategies are fully characterized by onelink deviation checks.
ON THE RELATION AMONG SOME DEFINITIONS OF STRATEGIC STABILITY
, 2001
"... In this paper we examine a number of different definitions of strategic stability and the relations among them. In particular, we show that the stability requirement given by Hillas (1990) is weaker than the requirements involved in the various definitions of stability in Mertens ’ reformulation of ..."
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In this paper we examine a number of different definitions of strategic stability and the relations among them. In particular, we show that the stability requirement given by Hillas (1990) is weaker than the requirements involved in the various definitions of stability in Mertens ’ reformulation of stability (Mertens 1989, 1991). To this end, we introduce a new definition of stability and show that it is equivalent to (a variant of) the definition given by Hillas (1990). We also use the equivalence of our new definition with the definition of Hillas to provide correct proofs of some of the results that were originally claimed (and incorrectly “proved”) in Hillas (1990).
The computational complexity of trembling hand perfection and other equilibrium refinments
 In SAGT
, 2010
"... Abstract. The king of refinements of Nash equilibrium is trembling hand perfection. We show that it is NPhard and SqrtSumhard to decide if a given pure strategy Nash equilibrium of a given threeplayer game in strategic form with integer payoffs is trembling hand perfect. Analogous results are sh ..."
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Abstract. The king of refinements of Nash equilibrium is trembling hand perfection. We show that it is NPhard and SqrtSumhard to decide if a given pure strategy Nash equilibrium of a given threeplayer game in strategic form with integer payoffs is trembling hand perfect. Analogous results are shown for a number of other solution concepts, including proper equilibrium, (the strategy part of) sequential equilibrium, quasiperfect equilibrium and CURB. The proofs all use a reduction from the problem of comparing the minmax value of a threeplayer game in strategic form to a given rational number. This problem was previously shown to be NPhard by Borgs et al., while a SqrtSum hardness result is given in this paper. The latter proof yields bounds on the algebraic degree of the minmax value of a threeplayer game that may be of independent interest. 1
Metastable Equilibria
, 2008
"... Metastability is a refinement of the Nash equilibria of a game derived from two conditions: embedding combines behavioral axioms called invariance and smallworlds, and continuity requires games with nearby best replies to have nearby equilibria. These conditions imply that a connected set of Nash e ..."
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Cited by 6 (4 self)
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Metastability is a refinement of the Nash equilibria of a game derived from two conditions: embedding combines behavioral axioms called invariance and smallworlds, and continuity requires games with nearby best replies to have nearby equilibria. These conditions imply that a connected set of Nash equilibria is metastable if it is arbitrarily close to an equilibrium of every sufficiently small perturbation of the bestreply correspondence of every game in which the given game is embedded as an independent subgame. Metastability satisfies the same decisiontheoretic properties as Mertens’ stronger refinement called stability. Metastability is characterized by a strong form of homotopic essentiality of the projection map from a neighborhood in the graph of equilibria over the space of strategy perturbations. Mertens’ definition differs by imposing homological essentiality, which implies a version of smallworlds satisfying a stronger decomposition property. Mertens’ stability and metastability select the same outcomes of generic extensiveform games.
ON FORWARD INDUCTION
, 2007
"... We examine Hillas and Kohlberg’s conjecture that invariance to the addition of payoffredundant strategies implies that a backward induction outcome survives deletion of strategies that are inferior replies to all equilibria with the same outcome. That is, invariance and backward induction imply for ..."
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We examine Hillas and Kohlberg’s conjecture that invariance to the addition of payoffredundant strategies implies that a backward induction outcome survives deletion of strategies that are inferior replies to all equilibria with the same outcome. That is, invariance and backward induction imply forward induction. Although it suffices in simple games to interpret backward induction as a subgameperfect or sequential equilibrium, to obtain general theorems we use a quasiperfect equilibrium, viz. a sequential equilibrium in strategies that are admissible continuations from each information set. Using this version of backward induction, we prove the HillasKohlberg conjecture for twoplayer extensiveform games with perfect recall. We also prove an analogous theorem for general games by interpreting backward induction as a proper equilibrium, since a proper equilibrium is equivalent to a quasiperfect equilibrium of each extensive form with the same normal form, provided beliefs are justified by perturbations invariant to inessential transformations of the extensive form. For a twoplayer game we prove that if a set of equilibria includes a proper equilibrium of every game with the same reduced normal form then it satisfies forward induction, i.e. it includes a proper equilibrium of the game after deleting strategies that are inferior replies to all equilibria in the set. We invoke slightly stronger versions of invariance and properness to handle nonlinearities in an Nplayer game.